Signal Representation/Analytical Signal and its Spectral Function - Revision history

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← Older revision		Revision as of 15:48, 19 June 2023
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	[[Aufgaben:Exercise 4.4: Vector Diagram for DSB-AM\|Exercise 4.4: Vector Diagram for DSB-AM]]		[[Aufgaben:Exercise 4.4: Vector Diagram for DSB-AM\|Exercise 4.4: Vector Diagram for DSB-AM]]

−	[[Aufgaben:Exercise 4.4Z: Vector Diagram for DSB-AM\|Exercise 4.4Z: Vector Diagram for ~~DSB~~-AM]]	+	[[Aufgaben:Exercise 4.4Z: Vector Diagram for DSB-AM\|Exercise 4.4Z: Vector Diagram for SSB-AM]]

@@ Line 24: / Line 24: @@
 From the graphic you can see the calculation rule for&nbsp; $X_+(f)$:&nbsp; The actual band-pass spectrum&nbsp; $X(f)$&nbsp; will
 *be doubled at the positive frequencies, and
 *set to zero at the negative frequencies.}}
 <br clear=all>

@@ Line 115: / Line 115: @@
 ==Pointer diagram representation of the harmonic oscillation==
 <br>
-The spectral function&nbsp; $X(f)$&nbsp; of a harmonic oscillation&nbsp; $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$&nbsp; consists of two Dirac delta functions at the frequencies
+The spectral function&nbsp; $X(f)$&nbsp; of a harmonic oscillation&nbsp; $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$&nbsp; consists of two Dirac delta functions at frequencies
-* $+f_{\rm T}$&nbsp; with the complex weight &nbsp; $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
+* $+f_{\rm T}$&nbsp; with complex weight &nbsp; $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
-* $-f_{\rm T}$&nbsp; with the complex weight &nbsp; $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.
@@ Line 125: / Line 126: @@
 T}) .$$
-The corresponding time function is obtained by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|$\text{Shifting Theorem}$]]:
+The corresponding time function is obtained by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|&raquo;Shifting Theorem&laquo;]]:
 :$$x_+(t) = A \cdot {\rm e}^{\hspace{0.05cm} {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}( 2 \pi f_{\rm T} t
@@ Line 132: / Line 133: @@
 This equation describes a rotating pointer with constant angular velocity&nbsp; $\omega_{\rm T} = 2\pi f_{\rm T}$.
-In the following, we will also refer to the time course of an analytical and frequency-discrete  signal&nbsp; $x_+(t)$&nbsp; as&nbsp; $\text{Pointer Diagram}$.
+In the following,&nbsp; we will also refer to the time course of an analytical and frequency-discrete  signal&nbsp; $x_+(t)$&nbsp; as&nbsp; &raquo;'''pointer diagram'''&laquo;.
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; For illustrative reasons the coordinate system here is rotated (real part upwards, imaginary part to the left), contrary to the usual representation by&nbsp; $90^\circ$.
+$\text{Example 3:}$&nbsp; For illustrative reasons the coordinate system here is rotated&nbsp; $($real part upwards,&nbsp; imaginary part to the left$)$,&nbsp; contrary to the usual representation by&nbsp; $90^\circ$.
 [[File:P_ID712__Sig_T_4_2_S3.png|right|frame|Pointer diagram of a harmonic oscillation]]
 On the basis of this diagram the following statements are possible:
-*At the start time&nbsp; $t = 0$&nbsp; the pointer of length&nbsp; $A$&nbsp; (amplitude) lies with angle&nbsp; $-\varphi$&nbsp; in the complex plane.&nbsp; In the drawn example,&nbsp; $\varphi = 45^\circ$.
+*At the start time&nbsp; $t = 0$&nbsp; the pointer of length&nbsp; $A$&nbsp; $($amplitude$)$&nbsp; lies with angle&nbsp; $-\varphi$&nbsp; in the complex plane.&nbsp; In the drawn example,&nbsp; $\varphi = 45^\circ$.
-*For the times&nbsp; $t > 0$&nbsp; the pointer rotates with constant angular velocity (circular frequency)&nbsp; $\omega_{\rm T}$&nbsp; in mathematically positive direction, i.e. counterclockwise.
-*The top of the pointer thus always lies on a circle with radius&nbsp; $A$&nbsp; and requires exactly the time&nbsp; $T_0$, i.e. the period duration of the harmonic oscillation&nbsp; $x(t)$&nbsp; for one rotation.
+*For times&nbsp; $t > 0$&nbsp; the pointer rotates with constant angular velocity&nbsp; $($circular frequency$)$&nbsp; $\omega_{\rm T}$&nbsp; in mathematically positive direction,&nbsp; i.e. counterclockwise.
 *The projection of the analytical signal&nbsp; $x_+(t)$&nbsp; onto the real axis, marked by red dots, provides the instantaneous values of&nbsp; $x(t)$.}}
@@ Line 148: / Line 152: @@
 ==Pointer diagram  of a sum of harmonic oscillations==
 <br>
-For further description we assume the following spectrum for the analytical signal:
+For the further description we assume the following spectrum for the analytical signal:
 [[File:P_ID715__Sig_T_4_2_S4.png|right|frame|Pointer diagram  of a sum of three oscillations]]
@@ Line 155: / Line 159: @@
 \varphi_i}\cdot\delta (f - f_{i}) .$$
-The left graphic shows such a spectrum for the example&nbsp; $I = 3$.&nbsp; If one chooses&nbsp; $I$&nbsp; relatively large and the distance between adjacent spectral lines correspondingly small, then (frequency&ndash;) continuous spectral functions&nbsp; $X_+(f)$&nbsp; can also be approximated with the above equation.
+#The left graphic shows such a spectrum for the example&nbsp; $I = 3$.&nbsp;
 In the right graphic the corresponding time function is indicated.&nbsp; This is in general:
@@ Line 163: / Line 169: @@
 To note about this graphic:
-*The sketch shows the initial position of the pointers at the start time&nbsp; $t = 0$&nbsp; corresponding to the amplitudes&nbsp; $A_i$&nbsp; and the phase positions&nbsp; $\varphi_i$.
+*The sketch shows the initial position of the pointers at start time&nbsp; $t = 0$&nbsp; corresponding to the amplitudes&nbsp; $A_i$&nbsp; and the phase positions&nbsp; $\varphi_i$.
 *The tip of the resulting pointer compound is marked by the violet cross.&nbsp; One obtains by vectorial addition of the three individual pointers for the time&nbsp; $t = 0$:
 :$$x_+(t= 0) = \big [1 \cdot \cos(60^\circ) - 1  \cdot {\rm j} \cdot \sin(60^\circ) \big ]+ 2 \cdot \cos(0^\circ)+1 \cdot \cos(180^\circ) = 1.500 - {\rm j} \cdot 0.866.$$
-*For times&nbsp; $t > 0$&nbsp; the three pointers rotate at different angular velocities&nbsp; $\omega_i = 2\pi f_i$.&nbsp; The red pointer rotates faster than the green one, but slower than the blue one.
+*For times&nbsp; $t > 0$&nbsp; the three pointers rotate at different angular velocities&nbsp; $\omega_i = 2\pi f_i$.&nbsp; The red pointer rotates faster than the green one,&nbsp; but slower than the blue one.
 *Since all pointers rotate counterclockwise, the resulting pointer&nbsp; $x_+(t)$&nbsp; will also tend to move in this direction.&nbsp;
 *At time&nbsp; $t = 1\,&micro;\text {s}$&nbsp; the tip of the resulting pointer for the given parameter values is
@@ Line 186: / Line 195: @@
-The interactive applet&nbsp; [[Applets:Physical_Signal_%26_Analytic_Signal|"Physical Signal and Analytical Signal"]]&nbsp; illustrates&nbsp; $x_+(t)$&nbsp; for the sum of three harmonic oscillations.
+The interactive applet&nbsp; [[Applets:Physical_Signal_%26_Analytic_Signal|&raquo;Physical Signal and Analytical Signal&laquo;]]&nbsp; illustrates&nbsp; $x_+(t)$&nbsp; for the sum of three harmonic oscillations.
 ==Exercises for the chapter==

@@ Line 12: / Line 12: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp; The&nbsp; $\text{analytical signal}$&nbsp; $x_+(t)$&nbsp; belonging to the physical signal&nbsp; $x(t)$&nbsp; is that time function, whose spectrum fulfills the following property:
-[[File:Sig_T_4_2_S1a_Version2.png|right|frame|Analytical signal in the frequency domain '''KORREKTUR''']]
+[[File:EN_Sig_T_4_2_S1a.png|right|frame|Analytical signal in the frequency domain]]
 :$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
 X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

@@ Line 117: / Line 117: @@
 ==Pointer diagram representation of the harmonic oscillation==
 <br>
-The spectral function&nbsp; $X(f)$&nbsp; of a harmonic oscillation&nbsp; $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$&nbsp; consists of two Dirac '''KORREKTUR: delta''' functions at the frequencies
+The spectral function&nbsp; $X(f)$&nbsp; of a harmonic oscillation&nbsp; $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$&nbsp; consists of two Dirac delta functions at the frequencies
 * $+f_{\rm T}$&nbsp; with the complex weight &nbsp; $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
 * $-f_{\rm T}$&nbsp; with the complex weight &nbsp; $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.
-Thus, the spectrum of the analytical signal is&nbsp; $($without the Dirac '''KORREKTUR: delta''' function at the frequency&nbsp; $f =-f_{\rm T})$:
+Thus, the spectrum of the analytical signal is&nbsp; $($without the Dirac delta function at the frequency&nbsp; $f =-f_{\rm T})$:
 :$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm

@@ Line 1: / Line 1: @@
 {{Header
-|Untermenü=Bandpass Signals
+|Untermenü=Band-Pass Signals
 |Vorherige Seite=Differences and Similarities of LP and BP Signals
 |Nächste Seite=Equivalent Low-Pass Signal and Its Spectral Function

@@ Line 75: / Line 75: @@
 ==Representation with Hilbert transform==
 <br>
-At this point it is necessary to briefly discuss a further spectral transformation, which is dealt thoroughly in the book [[Linear_and_Time_Invariant_Systems/Folgerungen_aus_dem_Zuordnungssatz#Hilbert.E2.80.93Transformation|"Linear and Time-invariant Systems"]]&nbsp;.
+At this point it is necessary to briefly discuss a further spectral transformation, which is dealt thoroughly in the book [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Hilbert_transform|"Linear and Time-invariant Systems"]]&nbsp;.
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp; For the&nbsp; $\text{Hilbert transform}$&nbsp; $ {\rm H}\left\{x(t)\right\}$&nbsp; of a time function&nbsp; $x(t)$&nbsp; applies:

@@ Line 6: / Line 6: @@
 }}
-==Definition in the Frequency Domain==
+==Definition in the frequency domain==
 <br>
 We consider a real band-pass signal&nbsp; $x(t)$&nbsp; with the corresponding band-pass spectrum&nbsp; $X(f)$, which has an even real and an odd imaginary part with respect to the frequency zero point.&nbsp; It is assumed that the carrier frequency&nbsp; $f_{\rm T}$&nbsp; is much larger than the bandwidth of the band-pass signal&nbsp; $x(t)$.